When I read the Ricci Soliton geometric meaning, I get stuck in the plugging in the Ricci flow as picture below.I don't know how to plug in it,in my opinion, Ricci flow is $\partial_tg_{ij}=-2R_{ij}$. Whether it mean $\mathcal L_{X_P}g_{ij}+2R_{ij}+2\lambda g_{ij}=0$ ?If so , how to get Ricci soliton euqation ?

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Suppose $g(t)=\sigma(t)\phi_t^*g_0$ solves the equation $$\displaystyle\frac{\partial}{\partial t}g=-2Ric(g).$$

Since our expression for $g(t)$ solves this equation we can plug it into the left side of the equation and set $t=0$: $$-2Ric(g_0)=\displaystyle\frac{\partial}{\partial t}g\bigg{|}_{t=0}=\displaystyle\frac{\partial}{\partial t}(\sigma(t)\phi_t^*g_0)\bigg{|}_{t=0}=\sigma'(0)g_0+\mathcal{L}_Xg_0$$

If you define $\sigma'(0)=2\lambda$ you have the Ricci soliton equation.


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